Heterogeneous risk attitudes and waves of infection

Abstract

Many countries have experienced multiple waves of infection during the COVID-19 pandemic. We propose a novel but parsimonious extension of the SIR model, a CSIR model, that can endogenously generate waves. In the model, cautious individuals take appropriate prevention measures against the virus and are not exposed to infection risk. Incautious individuals do not take any measures and are susceptible to the risk of infection. Depending on the size of incautious and susceptible population, some cautious people lower their guard and become incautious-thus susceptible to the virus. When the virus spreads sufficiently, the population reaches "temporary" herd immunity and infection subsides thereafter. Yet, the inflow from the cautious to the susceptible eventually expands the susceptible population and leads to the next wave. We also show that the CSIR model is isomorphic to the SIR model with time-varying parameters.

Fig 1. Weekly number of new COVID-19 cases between January 2020 and November 2021.

(Source: WHO Coronavirus Dashboard).

Fig 2. Time paths of each population in the SIR model with γ = 0.05 and β = 0.1 (blue) and β = 0.15 (red).

Fig 3. Phase diagram of the SIR model with β = 0.1 and γ = 0.05.

Fig 4. Flow chart of SIR and CSIR models.

Fig 5. Time paths of C, S, I and R with α = 0.007, β = 1, and γ = 0.05.

Fig 6. Phase diagrams of a CSIR model.

Fig 7. Contour plot for the number of waves for T = 2000 with different values of α and γ.

Initial values are C(0) = 0.95, S(0) = 0.0495, I(0) = 0.0005, R(0) = 0 and β = 1.

Fig 8. Initial population of the cautious C(0) and the number of waves.

Fig 9. Phase diagram of S and K=IC.

Fig 10. Steady-state paths of the CSIR model when I0C0=αβ and S0=γα+β.

Fig 11. Time path of the CSIR model with β = 1 (blue) and β = 1.5 (red).

Fig 12. The paths of the infectious and recovered in CSIR and SIR models.